Suppose that $\bsV$ is a finite dimensional real Euclidean space, $M\subset \bsV$ is a smooth compact submanifold of dimension $m$ and codimension $r$ and we set

$$ N:=\dim \bsV=m+r. $$

For any nonnegative integer $c\leq \dim \bsV$ we denote by $\Graff^c(\bsV)$ the Grassmannian of affine subspaces of $\bsV$ of codimension $c$, by $\Gr^c(\bsV)$ the Grassmannian of codimension $c$ vector subspaces of $\bsV$. We set $\Gr_k(\bsV):=\Gr^{N-k}(\bsV)$.

The codimension $c$ Radon transform of a smooth function $f: M\to \bR$ is a function

Observe that $\eI^c(V)\to \bsV$ is a smooth fiber bundle with fiber $\Gr^r(\bsV)$. In particular, $\eI^c(M)\to M$ is the bundle obtained by restricting to the submanifold $M$. Its fiber is also $\Gr^c(\bsV)$.

The Grassmannain $\Gr^c(\bsV)$ is equipped with a canonical $O(\bsV)$-invariant metric with volume density $|d\gamma^c_\bsV|$ with total volume $\newcommand{\sbinom}[2]{\genfrac{[}{]}{0pt}{}{#1}{#2}}$

$$\int_{\Gr^c(\bsV)} |d\gamma_\bsV^c(L)|=\sbinom{N}{c}, $$

where $\sbinom{N}{c}$ is defined in equation (9.1.66) of the Lectures.

Now observe that we have a natural projection $\pi: \Graff^c(\bsV)\to \Gr^c(\bsV)$ that associates to each affine plane its translate through the origin. A plane $S\in\Graff^c(\bsV)$ intersects the orthogonal complement of $\pi(S)$ in a unique point $C(S)=S\cap \pi(S)^\perp$. We obtain a an embeding

$$ \Gr^c(\bsV)\ni S\mapsto \bigl(\;C(S), \pi(S)\;\bigr)\in \bsV\times\Gr^c(\bsV),\;\;C(S)\perp \pi(S), $$
$\newcommand{\eQ}{\mathfrak{Q}}$
and we will regard $\Graff^c(\bsV)$ as a submanifold of $\bsV\times \Gr^c(\bsV)$. As such, it becomes the total space of a vector bundle $\eQ_c\to\Gr^c(\bsV)$, in fact a subbundle of the trivial bundle $\bsV\times \Gr^c(\bsV)\to\Gr^c(\bsV)$. The orthogonal complement $\eQ_c^\perp$ of this bundle is the tautological vector bundle $\newcommand{\eU}{\mathscr{U}}$ ${\eU}^c\to\Gr^c(\bsV)$. In particular

$$\dim\Gr^c(\bsV)= c(N-c)+ c. $$

Along $\Graff^c(\bsV)$ we have a canonical vector bundle, the vertical bundle $VT\Graff^c(\bsV)\subset T\Graff^c(\bsV)$ consisting of the kernels of $d\pi$, i.e., vectors tangent to the fibers of $\pi$. The vertical bundle is equipped with a natural density $|d\bv|_c$ which when restricted to a fiber of $\pi^{-1}(L)$ induces the natural volume form on the fiber $L^\perp$ viewed as a vector subspace of $\bsV$. As in Section 9.1.3 of the Lectureswe define a product density $|d\tilde{\gamma}^c|=|d\tilde{\gamma}_\bsV^c|$ on $ \Graff^c(\bsV)$,

$$|d\tilde{\gamma}_\bsV^c|= |d\bv|_c\times \pi^*|d\gamma_\bsV^{c}| $$

Alternatively, the vector bundle $\eQ_c$, as a subbundle of the trivial bundle $\bsV\times \Gr^c(\bsV)\to\Gr^c(\bsV)$ is equipped with a natural metric connection. The horizontal subbundle $HT\eQ_c\subset T\eQ_c$ is isomorphic to $\pi^* T\Gr^c(\bsV)$ and thus comes equipped with a natural metric. The vertical subbundle $VT\eQ_c=VT\Graff^c(\bsV)$ is also equipped with a natural metric and in this fashion we obtain a metric on $\Graff^c(\bsV)=\eQ_c$. The density $|d\tilde{\gamma}^c_\bsV|$ is the volume density defined by this metric.

Suppose now that $c\leq m=\dim M$. We denote by $\Graff^c_*(M)$ the subset of $\Graff^c(M)$ consisting of affine planes that intersect $M$ transversally. This is an open subset of $\Graff^c(M)$. The condition $c\leq m$ implies that this set is nonempty. (For $c=1$ this follows from the fact that the restriction to $M$ of a generic linear function is a Morse function. Then look at iterated slicing by hyperplanes.)

Set

$$ \eI^c_*(M)= \rho^{-1}\bigl(\;\Graff^c_*(M)\;\bigr)\subset \eI_M $$

The fiber of $\rho:\eI_*^c(M)\to \Graff^c_*(M)$ over $S\in \Graff^c_*(M)$ is the submanifold $S\cap M$ which is equipped with a metric density. We obtain a density on $\eI^c_*(M)$

For any vector subspace $U\subset \bsV)$ we denote by $\Gr^c(\bsV)_U$ the set consisiting of subspaces $L\in\Gr^c(\bsV)$ that intersect $U$ transversely.

We now want to integrate $\lambda^*(f)$ along the fibers of $\lambda :\eI^c_*(M)\to M$. For any vector subspace $U\subset \bsV)$ we denote by $\Gr^c(\bsV)_U$
the set consisting of subspaces $L\in\Gr^c(\bsV)$ that intersect $U$ transversely.

The fiber of this map over a point $x\in M$ is an open subset of $x+\Gr^c(\bsV)_{T_xM}\subset \Graff^c(\bsV)$ with negligible complement. The density $|d\nu^c_M|$ on $\eI^c_*(M)$ induces a density

The density $|d\nu^c(x)|$ is the restriction of a density $|d\bar{\nu}^c_x|$ on $\Gr^c(\bsV)_{T_xM}$. In fact, a reasoning similar to the one in the proof of Lemma 9.3.21 in the Lectures implies that for any $U\in\Gr_m(\bsV)$ there exists a canonical density $|d\bar{\nu}^c_U|$ on $\Gr^c(\bsV)_U$ such that

To find the constant $Z(N,m,c)$ we choose $M$ and $f$ judiciously. We let $M=\Sigma^m$, the unit $m$-dimensional sphere contained in some $(m+1)$-dimensional subspace of $\bsV$. Then, we let $f\equiv 1$. We deduce from (\ref{5}) that

Observe that $\widehat{f}$ has compact support. Indeed, if the support of $f$ is contained in a ball of radius $R$, then for any affine plane $S\in \Graff^c(\bsV)$ such that ${\rm dist}\,(0,S)>R$ we have $\widehat{f}(S)=0$.

Arguing as above, with $M=\bsV$ we observe that $\Graff^c_*(\bsV)=\Graff^c(\bsV)$ and we obtain as in (\ref{nu}) a density $|\nu^c_\bsV|$ on $\eI^c_*(\bsV)=\eI^c(\bsV)$. Denote by $\rho_*\Phi |d\nu^c_\bsV|$ the pushfoward of the density $\Phi|d\nu^c_\bsV$. It is a density on $\Graff^c(\bsV)$ and we have the Fubini formula (coarea formula)

and in this case we reobtain (\ref{5}) in the special case $M=\bsV$. The equality (\ref{d}) is important for another reason.

Denote by $C_0^{-\infty}(\bsV)$ the space of generalized functions with compact supports, then we can extend the Radon transform to such objects. If $u\in C_0^{-\infty}(\bsV)$ then we define its Radon transform $\widehat{u}$ to be the compactly supported generalized density on $\Graff^c(\bsV)$ defined by the equality

If $M$ is a compact submanifold of $\bsV$, then we get a Dirac-type generalized function $\delta_M$ on $\bsV$ defined by integration along $M$ with respect to the volume density on $M$ determined by the induced metric. Then